Which of the following triangles are isosceles as well as obtuse-angled triangles?
Fig 1 and Fig 3 only
An obtuse angled triangle is the triangle in which one of the angles is greater than 90∘.
An isosceles triangle is the triangle in which two sides are equal.
1. Fig 1:
ΔPQR is isosceles [∵PQ=PR]
⇒∠Q=∠R
[∵ angles opposite to equal sides of a triangle are equal]
∠P+∠Q+∠R =180∘
[angle sum property of a triangle]
∠P+25∘ +25∘ =180∘
⇒∠P=180∘ −50∘=130∘
ΔPQR is an obtuse angled triangle as one of the angles measures 130°.
2. Fig 2:
ΔABC is isosceles [∵AB=AC]
Similarly as above, we can find the angles of this triangle.
∠A=35∘,∠B=∠C=72.5∘
Since all angles are less than 90∘, ΔABC is an acute angled triangle.
3. Fig 3:
ΔXYZ is an isosceles as well as an obtuse angled triangle as angle Y measures 110°.
4. Fig 4:
ΔMNO is an isosceles as well as a right angled triangle.
Hence, only Fig 1 and Fig 3 are isosceles as well as obtuse angled triangles.